Re: Iteration

Hi Bob and Still Learning;

If you try that one you will get a surprise.

If we start with an initial condition of 1 we get the following diagram called a cobweb. See fig 1.

The arrow shows where the point ends up after 5 iterations. Notice it is getting further away from the root. We say it being repelled. To show that not even a closer guess will help, the second figure shows an initial condition of 1.3. After 12 iterations the arrow shows where it ended up. Further away from the root then when it started.

So although there zillions of iterative forms not all of them converge on a root. A little calculus is needed to know when convergence is assured and how fast it will be.

Try this one on your own,

In mathematics, you don't understand things. You just get used to them.

Re: Iteration

Iteration is more robust than the classical methods taught. It is a general method to solve any type of equation. You can use any initial condition but not all of them will converge. You would of course try to get a good initial guess for a simple system like this.

In real life the number of equations is in the hundreds or thousands with that many variables. Intelligent guessing is out unless you can plot in thousands of dimensions. We would just pick 1 or 0 for all the initial conditions.

Wow,that converges at 1.92318... And 0.54894...,why?

It does not converge it will keep being repelled by the root, going further and further away.

In mathematics, you don't understand things. You just get used to them.

Re: Iteration

Re: Iteration

Usually it is trial and error. To find other answers you try different initial conditions. For that cubic you would now try x1 = 1 + i maybe, since you know the other two answers are in the complex plane.

In mathematics, you don't understand things. You just get used to them.

Re: Iteration

I think he wants to know how to know which x to solve for to actually get the solutions that do not diverge...

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.